A216584 a(n) = A002426(n)*A000984(n); product of central trinomial coefficients and central binomial coefficients.
1, 2, 18, 140, 1330, 12852, 130284, 1348776, 14247090, 152618180, 1654120468, 18096447096, 199536967084, 2214714164600, 24720932068200, 277289164574640, 3123590583844530, 35318969120870820, 400692715550057700, 4559427798654821400, 52020436064931914580
Offset: 0
Keywords
Examples
L.g.f.: L(x) = 2*x + 18*x^2/2 + 140*x^3/3 + 1330*x^4/4 + 12852*x^5/5 + 130284*x^6/6 + ... where exp(L(x)) = 1 + 2*x + 11*x^2 + 66*x^3 + 485*x^4 + 3842*x^5 + 32712*x^6 + ... + A216585(n)*x^n/n + ... The central trinomial coefficients (A002426) begin: [1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, ...]; The central binomial coefficients (A000984) begin: [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, ...].
Links
- G. C. Greubel, Table of n, a(n) for n = 0..925
Programs
-
Mathematica
Table[Binomial[2*n, n]*Sum[ Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}], {n,0,50}] (* G. C. Greubel, Feb 27 2017 *) -
PARI
{a(n) = polcoeff((1+x+x^2)^n,n) * polcoeff((1+2*x+x^2)^n,n)} -
PARI
{a(n)=binomial(2*n,n)*sum(k=0,n\2,binomial(n,2*k)*binomial(2*k,k))} for(n=0,21,print1(a(n),", "))
Formula
a(n) = binomial(2*n, n) * Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k, k).
Logarithmic derivative of A216585, after ignoring initial term a(0).
a(n) = [x^n*y^n] ( 1 + (x + y)^2 + (x + y)^4 )^n. - Peter Bala, Feb 17 2020
G.f.: hypergeom([1/2, 1/2],[1],16*x/(1+4*x))/sqrt(1+4*x). - Mark van Hoeij, May 13 2025